Question: "Only one remains." Ryan signals to his brother from his hiding place. Matt nods in acknowledgement, spotting the last evil robot. " $34$ degrees." Matt signals back, informing Ryan of the angle he observed between Ryan and the robot. Ryan records this value on his diagram (shown below) and performs a calculation. Calibrating his laser cannon to the correct distance, he stands, aims, and fires. To what distance did Ryan calibrate his laser cannon? Do not round during your calculations. Round your final answer to the nearest meter.
Explanation: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $BC=d$. Because the interior angles of a triangle add to $180^\circ$, we know that $\angle B=26^\circ$. $A$ $B$ $C$ $34^\circ$ $120^\circ$ $26^\circ$ $146\text{ m}$ $d$ Since we are given one side length and all angle measures, we can use the law of sines. Using the law of sines $\begin{aligned} \dfrac{\sin(B)}{AC}&=\dfrac{\sin(A)}{BC}\\\\ \dfrac{\sin(26^\circ)}{146} &= \dfrac{\sin(34^\circ)}{d} \gray{\text{Substitute}} \\\\ d \cdot \sin(26^\circ) &= 146 \cdot \sin(34^\circ) \\\\ d &= \dfrac{146 \cdot \sin(34^\circ) }{\sin(26^\circ) } \\\\ d &\approx 186 \,\text{m} \end{aligned}$ The answer Ryan should calibrate his laser cannon to $186 \,\text{m}$.